Ultrametricity in the Edwards-Anderson Model

Contucci, Pierluigi; Giardinà, Cristian; Giberti, Claudio; Parisi, Giorgio; Vernia, Cecilia

doi:10.1103/physrevlett.99.057206

Cited by 40 publications

(77 citation statements)

References 22 publications

(37 reference statements)

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“…1). In [1], these data are used as a strong hint for the presence of a spin glass phase with an ultrametric structure while we obtain almost identical results in the 2d model where it is widely accepted that (i) there is no spin glass phase at any finite temperature, and that (ii) the transition is well-described by the droplet picture.…”

supporting

confidence: 54%

“…To conclude, we believe that although the data published in [1] may be compatible with the presence of an ultrametric structure, they are, however, not sufficient to dismiss the possibility that other models as, e.g., the droplet model might apply to the 3d EA spin glass. We also want to emphasize the need of comparing with simple models in order to validate the conclusions reached in large-scale numerical experiments, especially in the context of spin glasses where simulations and their interpretation are known to be difficult.…”

mentioning

confidence: 77%

“…In a recent interesting Letter Contucci et al [1] have investigated several properties of the three-dimensional (3d) Edwards-Anderson (EA) Ising spin glass. They claim to have found strong numerical evidence for the presence of a complex ultrametric structure similar to the one described by the celebrated replica symmetry breaking (RSB) solution of the mean field model [2].…”

mentioning

confidence: 99%

“…The authors of [1] state that the droplet model, where only one state and its reversal symmetric exist in the thermodynamic limit, cannot satisfy non-trivial ultrametricity because its P (Q) is trivial so that only equilateral triangles will be observed for the link overlap. This is not quite true, as its P (Q) (for both link and site overlap) for finite sizes and temperatures can be far from showing a trivial structure.…”

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confidence: 99%

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Comment on “Ultrametricity in the Edwards-Anderson Model”

Jörg

Krząkała

2008

Phys. Rev. Lett.

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supporting

confidence: 54%

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confidence: 77%

mentioning

confidence: 99%

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confidence: 99%

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Comment on “Ultrametricity in the Edwards-Anderson Model”

Jörg

Krząkała

2008

Phys. Rev. Lett.

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“…Unfortunately, the existence of an UM phase structure for SR spin glasses is controversial, mainly because only small linear system sizes have been accessible so far. Recent results [11] suggest that SR systems are not UM, whereas other opinions exist [12,13,14,15]. Thus it is of paramount importance to test if SR spin glasses have an UM phase space.…”

mentioning

confidence: 99%

Ultrametricity and Clustering of States in Spin Glasses: A One-Dimensional View

Katzgraber

Hartmann

2009

Phys. Rev. Lett.

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We present results from Monte Carlo simulations to test for ultrametricity and clustering properties in spin-glass models. By using a one-dimensional Ising spin glass with random power-law interactions where the universality class of the model can be tuned by changing the power-law exponent, we find signatures of ultrametric behavior both in the mean-field and non-mean-field universality classes for large linear system sizes. Furthermore, we confirm the existence of nontrivial connected components in phase space via a clustering analysis of configurations. The nature of the spin-glass state is controversial and it is unclear if the mean-field replica symmetry breaking (RSB) picture [2], the droplet picture [7,8], or an intermediate phenomenological scenario dubbed as TNT [9,10] (for "trivial-nontrivial") describes the nature of the spin-glass state best. One way to settle the applicability of the RSB picture to short-range (SR) spin glasses is by testing if the phase space is UM. Unfortunately, the existence of an UM phase structure for SR spin glasses is controversial, mainly because only small linear system sizes have been accessible so far. Recent results [11] suggest that SR systems are not UM, whereas other opinions exist [12,13,14,15]. Thus it is of paramount importance to test if SR spin glasses have an UM phase space.In this work we approach the problem from a different angle: First, we use a one-dimensional (1D) Ising spin-glass with power-law interactions. The model has the advantage that large linear system sizes can be studied. Furthermore, by tuning the exponent of the power law, the universality class of the model can be tuned between a mean-field and a non-mean-field universality class. This allows us to test our analysis method on the mean-field SK model and then apply it to regions of phase space where the system is not mean-field like. We perform a clustering analysis of the data similar to the work of Hed et al.[11] to obtain nontrivial triangles in phase space and introduce a novel correlator which allows us to see an UM signature for low temperatures and delivers we expect SK-like infinite-range behaviour. For 1/2 < σ ≤ 2/3 we have mean-field (MF) behaviour corresponding to an effective space dimension d eff ≥ 6, whereas for 2/3 < σ ≤ 1 we have a long-range (non-MF) spin glass with a ordering temperature Tc > 0. Close to σ = 2/3 (vertical red line)no signal for high temperatures. Furthermore, we use a clustering analysis to search for connected components in phase space. The proposed method can be applied to any field of science to test for an UM structure of phase space, thus making the method generally applicable. Our results for low temperatures show that for this model the phase space has an UM signature and exhibits many phase-space components, the number growing with system size in the mean-field as well as non-mean-field case. This suggests that for large enough system sizes SR spin glasses at low enough temperatures might have an UM phase space structure.Model.-The Hamiltonian of the...

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